The Triangular Form of the Truncated Shift
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Decompositions of the model operator into spectral subspaces, which in one way or the other are analogous to the Spectral Theorem, will be the object of the next few Lectures. As we will see, the possibility of such a decomposition depends on special (and rather restrictive) requirements imposed on the operator. The classical theory of matrices indicates however that it can be of considerable use already to have a transformation of the operator in question to “triangular form”, i.e. representing it as a matrix operator with respect to a basis gotten by orthogonalization from the corresponding family of eigen and root vectors. We give here a continuous analogue of such a “triangularization” for an arbitrary truncated shift using decreasing chains of invariant subspaces.
KeywordsInvariant Subspace Blaschke Product Volterra Operator Pure Point Spectrum Spectral Mapping Theorem
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